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In this paper we study a maximal operator $ℳf$ related with the best $\varphi$ approximation by constants for a function $f\in L_{\text{loc}}^{{\varphi }^{\text{'}}}\left({ℝ}^n\right)$, where we denote by ${\varphi }^{\text{'}}$ derivative function of the $C^1$ convex function $\varphi$. We get a necessary and sufficient condition which assure strong inequalities of the type ${\int }_{{ℝ}^n}\theta \left(ℳ\right|f\left|\right)dx\le K{\int }_{{ℝ}^n}\theta \left(\right|f\left|\right)dx$, where $K$ is a constant independent of $f$. Some pointwise and mean convergence results are obtained. In the particular case $\varphi \left(t\right)=t^{p+1}$ we obtain several equivalent conditions on the functions $\theta$ that assures strong inequalities of this type....